This study will evaluate patients with blood cell cancers who are going to have an allogeneic (donor) blood stem cell transplant from a partially matched relative. The research study will test whether immune cells, called T cells, which come from the donor relative and are specially grown in the laboratory and then given back to the patient along with the stem cell transplant (T cell addback), can help the immune system recover faster after the transplant. As a safety measure, these T cells have been "programmed" with a "self-destruct switch" so that if, after they have been given to the patient, the T cells start to react against the tissues (called "graft versus host" disease, GVHD), the T cells can be destroyed.